The Justice of Proportions

Previously, I posted some proportional studies of male and female figures. Male HERE and female HERE.

People fall within a range of proportions. Men can range from 7:1 on the squat and robust end, to 9:1 on the attenuated end (this proportion is reserved for adolescent boys and delicately proportioned men) Women can range from 8:1 on the more full-figured side of the scale to 10:1 for adolescents and very ethereally proportioned girls. Why the importance? Because if you're trying to draw a matronly lady you'd proportion her at 8:1. A barrel chested farmer who has spent his life working in the fields would be 7:1. A vestal virgin would be 9 1/2 or 10:1, while the effeminate and lisping Alcibiades would probably be well represented by an epicene body that is 8 1/2 or even 9 heads high--it all depends on what you're trying to represent to the audience. One thing is clear... Alcibiades at 7:1 wouldn't be right.

All this doesn't take into account the proportion of children which never meet the mark of adults--a phenomenon found in the higher mammals known as allometric growth--they start out in life with a head 25% or more of their body length, and then gradually stretch, becoming overly attenuated at adolescence before they elastically snap back to the overall proportion of maturity they'll maintain until old age, at which time they head back to, well, 0:1.


What is being set up here is a set of 'keys' for architectural design. There are 5 basic keys in the ordering of architecture. In fact, they are called the 5 'Orders', and many of you just call them the different kinds of classical columns. They are, in order of robust and non-sculptural to attenuated and highly ornamented, Tuscan, Doric, Ionic, Corinthian, and Roman (sometimes called Composite). Above is a parallel of the orders I drew comparing the orders to each other, keeping the height static and the lower diameter, or modulus, variable. How are they related to the human figure? Because the classical orders, anthropomorphic in origin and symbol, share the same basic proportions as the range of human figures. The Tuscan is 7:1, the Doric 8:1, Ionic 9:1, while Corinthian and Roman are 10:1, but differ in the amount and delicateness of ornamentation. Note that along the scale there is no ornamentation on the Tuscan (who's heard of a country farmer wearing jewelry?) All the way up to highly ornamented in the effeminate range of Corinthian and Composite.

Now, back to why they're called 'Orders'. They are given this name because they give order, they modulate the style, proportion and ornamentation of, the architecture. This is most clearly understood when it comes to building typologies. While it would be fitting for a Theater, the function of which is delicate, entertaining and frivolous to be in the 'key' of Corinthian or Composite, it would be an altogether misleading impropriety for a bank to be of such an order, given that it needs to project the ideas of sturdiness, lastingness, and impenetrability. So too, when choosing the order for a certain church dedication: It would be fitting for a church dedicated to St. Paul to be ordered with the Doric order, as he was a strong, princely figure. But to choose Doric rather than Composite for a church dedicated to St. Therese of Lisieux would be to represent the girl rather not as a little flower but as a douglas fir.

Granted, this is modulation 101, and while architects can most certainly change 'keys' within one piece of art in the same way as Bach does in his music, we nevertheless have to learn the basics first, or you designers out there will totally screw up your façade, also known as façadomy. This is often why I'm so frequently losing my temper at idiotic uses of columnar elements in McMansions and the like. Contemporary architects are ruining a perfectly good and clear symbology when they slap up a Tuscan column here, a Corinthian there, an Ionic around the bathtub, just for the 'look at me I'm wealthy factor'. Heck, in the majority of designs, you don't even NEED a column for the Order to be present, but rather the proportional and ornamental principles ought be present in all features and done in a commodius manner, and never for a Mannerist commode, as one would find at Mike's Marbleopolis:

God is in the Details

When in the Course of human events it becomes necessary for
one people to dissolve the political bands which have connected
them with another and to assume among the powers of the earth,
the separate and equal station to which the Laws of Nature and of
Nature's God entitle them, a decent respect to the opinions of
mankind requires that they should declare the causes which impel
them to the separation.

Printed above is the first paragraph of the U.S. Declaration
of Independence. Select any one of the first 20 words. Count
the number of letters in that word and call that number n. Move ahead n words,
beginning with the word after your selected word. When you reach
that nth word, count its letters and move ahead as many words as
the new letter count. Continue in this manner, counting letters
and moving ahead words, until you stop on a word that's beyond
the fourth line.

On what word did you stop?

The Physics of Santa Claus

As the father of a curious 7 year old, as well as a 9 year old still in denial, I've been having a hard time overcoming these plain facts when confronted with the question of whether Santa really exists:

1. No known species of reindeer can fly. BUT there are an estimated 300,000 species of living organisms yet to be classified, and while most of these are insects and germs, this does not COMPLETELY rule out flying reindeer which only Santa has ever seen.
2. There are 2 billion children (persons under 18) in the world. BUT since Santa doesn't (appear) to directly handle the Muslim, Hindu, Jewish and Buddhist children, that reduces the workload to 15% of the total : 378 million according to Population Reference Bureau. At an average (census)rate of 3.5 children per household, that's 91.8 million homes. One presumes there's at least one good child in each.
3. Santa has 31 hours of Christmas to work with, thanks to the different time zones and the rotation of the earth, assuming he travels east to west(which seems logical). This works out to 822.6 visits per second. This is to say that for each Christian household with good children, Santa has 1/1000th of a second to park, hop out of the sleigh, jump down the chimney, fill the stockings, distribute the remaining presents under the tree, eat whatever snacks have been left, get back up the chimney, get back into the sleigh and move on to the next house. Assuming that each of these 91.8 million stops are evenly distributed around the earth (which, of course, we know to be false but for the purposes of our calculations we will accept), we are now talking about .78 miles per household, a total trip of 75-1/2 million miles, not counting stops to do what most of us must do at least once every 31 hours, plus feeding and etc. This means that Santa's sleigh is moving at 650 miles per second, 3,000 times the speed of sound. For purposes of comparison, the fastest man-made vehicle on earth, the Ulysses space probe, moves at a poky 27.4 miles per second : a conventional reindeer can run, tops, 15 miles per hour.
4. The payload on the sleigh adds another interesting element. Assuming that each child gets nothing more than a medium-sized lego set (2 pounds), the sleigh is carrying 321,300 tons, not counting Santa, who is invariably described as overweight. On land, conventional reindeer can pull no more than 300 pounds. Even granting that 'flying reindeer' (see point #1) could pull TEN TIMES the normal anoint, we cannot do the job with eight, or even nine. We need 214,200 reindeer. This increases the payload - not even counting the weight of the sleigh : to 353,430 tons. Again, for comparison : this is four times the weight of the Queen Elizabeth.
5. 353,000 tons traveling at 650 miles per second creates enormous air resistance : this will heat the reindeer up in the same fashion as spacecrafts re-entering the earth's atmosphere. The lead pair of reindeer will absorb 14.3 QUINTILLION joules of energy. Per second. Each. In short, they will burst into flame almost instantaneously, exposing the reindeer behind them, and create deafening sonic booms in their wake. The entire reindeer team will be vaporized within 4.26 thousandths of a second. Santa, meanwhile, will be subjected to centrifugal forces 17,500.06 times greater than gravity. A 250-pound Santa (which seems ludicrously slim) would be pinned to the back of his sleigh by 4,315,015 pounds of force.

Of course, Saint Nicholas could pull it off. But Santa? Apparently not.
Time to reinforce the symbolic nature of Ol' Saint Nick-the bishop, not the Coca-Cola Salesman.

8 1/2 Heads High

In my LAST POST, I showed that the male figure is generally 8 heads high. While the general rule of thumb for figure drawing is that the whole body is 8 heads high, male or female, I would take issue with this. Today we look at the proportions of women...and note the female figure is 8 1/2 to 9 heads high. (This one drawn at 8 1/2 to 1)


Of course, all people fall within a range of proportions. And these variances of proportion can carry different connotations, meanings...

8 Heads High

By Ray Gunner's request, and since I've not been writing regularly, but have been drawing instead, I'm going to post some of the sketches I've been working on. Lately, it's been proportional studies of the human figure, where I'm trying to learn the (almost always) simple whole number ratios that govern the relationship of parts of the body. In this case, the male figure is shown to be 8 heads tall (taking the figure's own head height as the modulus/unit)

Bar Stool Economics

Suppose that every day, ten men go out for beer and the bill for all ten comes to $100. If they paid their bill the way we pay our taxes, it would go something like this:

The first four men (the poorest) would pay nothing.

The fifth would pay $1.

The sixth would pay $3.

The seventh would pay $7.

The eighth would pay $12.

The ninth would pay $18.

The tenth man (the richest) would pay $59.

So, that's what they decided to do.

The ten men drank in the bar every day and seemed quite happy with the arrangement, until one day, the owner threw them a curve. "Since you are all such good customers," he said, "I'm going to reduce the cost of your daily beer by $20."Drinks for the ten now cost just $80.

The group still wanted to pay their bill the way we pay our taxes so the first four men were unaffected. They would still drink for free. But what about the other six men - the paying customers? How could they divide the $20 windfall so that everyone would get his 'fair share?' They realized that $20 divided by six is $3.33. But if they subtracted that from everybody's share, then the fifth man and the sixth man would each end up being paid to drink his beer. So, the bar owner suggested that it would be fair to reduce each man's bill by roughly the same amount, and he proceeded to work out the amounts each should pay.

And so:

The fifth man, like the first four, now paid nothing (100% savings).

The sixth now paid $2 instead of $3 (33%savings).

The seventh now pay $5 instead of $7 (28%savings).

The eighth now paid $9 instead of $12 (25% savings).

The ninth now paid $14 instead of $18 (22% savings).

The tenth now paid $49 instead of $59 (16% savings).

Each of the six was better off than before. And the f irst f our continued to drink for free. But once outside the restaurant, the men began to compare their savings.

"I only got a dollar out of the $20,"declared the sixth man. He pointed to the tenth man," but he got $10!"

"Yeah, that's right," exclaimed the fifth man. "I only saved a dollar, too. It's unfair that he got ten times more than I!"

"That's true!!" shouted the seventh man. "Why should he get $10 back when I got only two? The wealthy get all the breaks!"

"Wait a minute," yelled the first four men in unison. "We didn't get anything at all. The system exploits the poor!"

The nine men surrounded the tenth and beat him up.

The next night the tenth man didn't show up for drinks, so the nine sat down and had beers without him. But when it came time to pay the bill, they discovered something important. They didn't have nearly enough money between all of them to even come close to paying the bill!

And that, boys and girls, journalists and college professors, is how our tax system works. The people who pay the highest taxes get the most benefit from a tax reduction. Tax them too much, attack them for being wealthy, and they just may not show up anymore. In fact, they might start drinking overseas where the atmosphere is somewhat friendlier.


Hat tip to:
David R. Kamerschen, Ph.D.
Professor of Economics
University of Georgia

No More Reason to Diet?

Ok, well I had to drink a couple glasses of water to get it just right, but right now I weigh:

1.61803398874989484820 x 10¹ STONES.

So this is probably the only time in my life I could say with all honesty that I've got DIVINE PROPORTIONS. (And it's all thanks to taking extra portions)

Next up: Lean and Ex-creme Ration

Theory vs. Practice

In Mathematicks he was greater
Than Tycho Brahe or Erra Pater:
For he, by geometrick scale,
Could take the size of pots of ale;
Resolve, by sines and tangents, straight,
If bread or butter wanted weight;
And wisely tell what hour o' th' day
The clock does strike by algebrae.

-From Samuel Butler's Hudibras.

I'm Still On Cloud Nine

UPDATE: Her eyes did well up, and then they rolled. Epic Success.

I'm the kind of guy who gets really excited over the odometer rolling to a new number. Not only do I get excited when it clicks to a nice round multiple of 10000, but also fun numbers like 75757.5, or 111111 (I made a point of driving 11 mph when I hit that one. No. Seriously. I pulled to the shoulder of the PCH, which is...Hwy 1). Palindromes are always exciting: 98338.9, or 48384. But it's seeing my favorite numbers like π or e that gives me a unique pleasure, the all-time greatest being when the odometer in my Mustang read out the golden number Φ:


Notice that the digits span beyond the odometer to include the trip meter. A little bit of planning, and voila! An increased level of precision, which paints a more exacting picture of your nerdiness to your wife, so that she can use it to more accurately make fun of you.

But now I have one that I'm hoping won't generate mockery from my beauty, although she will certainly roll her eyes after she dries them.

Today is a special day. Tonight, at around, oh, about 9:09:09 PM, my wife and I will have been married for 9 years, 9 months, 9 days, 9 hours, 9 minutes and 9 seconds. So, as you're reading this we're on our way up to San Francisco for a surprise overnight nonce. Why go so far for this? Because there's a restaurant up there called The Nines. And we're gonna go there for dinner after we play some 9-Ball (9 games of it, for sure, not to mention we'll squeeze in 9 innings of listening to the Dodgers lose on the radio.). And waiting at our table will be nine roses. And because of the wonderful people there at the restaurant, Beethoven's 9th Symphony will be playing in the background (and "1999" and "99 luftballons"...)

So you see, I am compelled to go the whole nine yards, or my psyche will be...will be...DECIMATED. And who knows? Maybe nine months from now...

At any rate, your Ninecompoop loves ya, Maria!

Gravy Trains & Gravity Trains

Eoin Colfer, known for such distinguished intellectual masterpieces as "Eoin Colfer's Legend of the Worst Boy in the World", and "Going Potty" is going to be FINISHING UP Douglas Adam's 'The Hitchhiker's Guide To the Galaxy'.

I say that as if before today I knew who he was, or even care at this time. But I didn't and I don't. In fact, I think that only Douglas Adams could follow up a non-sequitur such as Douglas Adams. So, while I always wanted The Guide to be finished, I'm very unhappy The Guide will be done by this poser, because more than likely, it'll be done for by virtue of being a palimpsest . Then again, it's not like it should take a lot of effort. Write funny jibberish for 300 pages, occasionally poking fun of the eccentricities of British introverts. Easy money.

On the other hand, this kind of thing makes me very, very happy to be thinking about The Guide:



Cool, huh? Unfortunately, it takes 42 mins 12 seconds, not exactly 42 minutes like the video makes it seem, and this is only true for straight line tubes. Curved tubes can generate varying times, and depending on the tube shape, can get you from A to B in much less time than 42.2 minutes. Nevertheless, it really bugs me that the Gravity Train misses the murine 42 by 12 seconds. I would love a nice round 42, which would validate my geocentric tendencies via the relationship between Earth and the answer to life, the universe, and everything.

So check out my solution to the problem killing OCD sufferers everywhere:

WARNING: DUCK ABOUT TO BORE YOU (even more than usual)

From Newton's law of Gravity, we have:

F=GmM/r²

The density of the Earth is:

ρ=3M/4 π r³

Substituting, we get:

F=4 G π m ρ r/3

Gravity through mass being what delivers the force to drive the object throught the tube, and is dependent on the mass of the two objects and their distance apart. Assuming the mass of the Gravity Train car stays constant, we see that the only variable in the formula is density (ρ), so the greater the density, the greater the force.

Filling in the blanks of the arguement, we apply Newton's second law of motion:

F=ma

The greater the force, the greater the acceleration. The greater the acceleration, the greater the speed. The greater the speed, the less time. And believe me, I really, really want to shave off that 12 seconds. (Hehe, can you feel it? It's like a pinewood derby race...slight alterations shaving off split seconds here and there...woohoo!)

So, the currently mass of the Earth is 5.98 x10²⁴kg, and the current density is 5.52g/cm³: that number being what determined not only the 42 minutes but also the 12 evil seconds. If, the process of aging, Gaia can put on enough poundage around her hips and fat butt to measure in at 5.55g/cm³, she will have increased her density by enough to reduce the Gravity train time by the needed 12 seconds.

Now, the Earth is actually getting more massive day by day through the accumulation of matter from space, from the likes of asteroids, meteoroids, interplanetary dust and even solar wind, to the order of 10⁸kg of nickel/iron blends per year. The average density of that new matter is 7.0g/cm³. Because the density of the new stuff is more than the current stuff, the density of Earth is rising. But, because more stuff is accumulating, this also means that the earth grows in volume by 14286m³/year, and therefore slows down the rate of change of density:

ρ_new=M_o+∆M₁+∆M₂.../V_o+∆V₁+∆V₂...

or:

ρ_new=((M_o+x(∆M))/(V_o+x(∆V)))-M_o/V_o

Where M_o/V_o=5.52g/cm³, and the integer value of indices represents the number of years needed to reach 5.55g/cm³.

Applying this, we see Earth's density rises by around 2.11 x 10^-9g/cm³ per year. In order to satisfy the need to raise Earth's density by .03g/cm³, this will take a lot of time. 14,218,009 years and 6 months, to be exact. But then the Gravity Train will actually run at 42 minutes even, and I will finally be happy, if only for an instant. And that would be just gravy.

Hurricane Tally

Easily among my top ten favorite XKCD comics:

Eerie. Weird. Cool. Reminiscent.

THIS is just plain beautiful--although one could almost say 'eeray'. The patterns are quite striking, and make me wonder if there's a ideal packing density solution descriptive of the pattern they form en masse. Perhaps it's just conjecture, but maybe there's a dodecahedral KISSING PATTERN if you treat each ray as swimming in a spherical 'bubble' of space? Yeah, at least a Kepler conjecture.


My wife and I had an experience like this when on a trip out to the Channel Islands 12 years ago. Our boat was swarmed by a pod of Dolphins, estimated by our whale watching guide to be around 10,000 strong: easily the largest pod he'd ever seen. Whatever the true number, as far as you could see they were leaping out of the water. 100s were in front of and to each side of the boat, with 30 in the air at any one moment. It was breathtaking.

Mega Cubes

Finally!!! After 3 years of waiting since Mr. Verdes announced that he found a way to mass produce them, the 6x6x6 and 7x7x7 cubes are out for sale! You may not share in my excitement, but to make the 7x was quite a feat of engineering.

In order to show the solution complexity of these cubes, take a look at the number of permutations (the number of possible ways to mix-up the cube, and therefore the number of possible solutions). Making some assumptions about the symmetricality of the pieces of the puzzle, I'm pretty confident that I correctly calculate the number of possible permutations as:

8! x 3⁷ x 24!⁴ x 48!³ x 12! x 2¹⁰ = 2.14 x 10²³⁷
4!²⁴ x 8!⁶

To put the size of this number into perspective, it's generally accepted that there are 'only' an estimated 4 x 10⁸¹ atoms in the entire universe. That's more than a tripleplex order of magnitude. Wow. Can you feel the excitement? Or, in another fantastically incomprehensible comparison between two ridiculously large numbers, our big number is within 1000 times of A SHANNON NUMBER of Shannon Numbers. (1 x 10²⁴⁰). Ha. Chess has got nothing on this big momma of cubes. Bear in mind that the original 3x3x3 Rubik's cube (that you nevertheless still peeled the stickers off of), has only a puny 4.3 x 10¹⁹ possible permutations. Pshaw. A THREE YEAR OLD can solve that one, and teenagers can solve it in LESS THAN 10 SECONDS. That's no competition for the 7x7x7.

And because I just know you're still with me, and so eager for more gratuitous quantifying, I'll go on.

So, there are two hundred fourteen septseptuagintillion possible ways to the mix up the 7x7x7. This is written out as:
214,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

And the whole point is to find the one correct solution.

Incidentally, if anyone feels so inclined to chip in, I would really, really like one of each, but my wife would kill me if I spent $80 on another cube. But, if you'd like something in return other than a daily dose of punishing complaints on your nth favorite blog, I'd be willing to trade: would you be interested in my drawing a PORTRAIT of you or your kid in exchange for a cube? That's a pretty good deal for a rubik's cube... (Does that sound desperate enough?)

.Verily A Gorgeous Numeration Day

I wish THIS would have shown up on Pi day. But, today's just as beautiful.

OK, I'm Fibbing. Today is much, much more beautiful.

Happy Phi Day everyone! It's a Phine day indeed!

(In honor of Fr. Barry, I'll give 1,123,581,321 WLIIA points to whoever can guess what's up with the title.)

Antarctica: The Sluttiest Continent on Earth

Ok, It's been well documented that many teenagers become promiscuous because of boredom. In fact, this is true for adults as well, and the fact isn't limited to the sexual appetite. 'An idle mind is the Devil's workshop.'. Now, of the major boring places in the world, Antarctica in the winter must be right up there at the top. But even so, I'm still shocked to hear that A SHIPMENT OF 16,500 CONDOMS was just delivered to the coldest continent. Apparently things get pretty hot down there in the winter amidst all that boredom.

But wait. There are only 124 people in Antarctica during the winter. A quick search on google yielded an average female/male ratio at 1:3 yielding a maximum of 31 mating pairs. (Logic is confined to the natural) Given that shipments will resume in mid September, those 16,500 condoms have to outlast the lust for about 100 days.

That's 5.32 unit units per each mating pair each and every day.

Huh?

Seriously. That's ridiculous. Bring a book or some crossword puzzles.

An Argument With Gaping Black Holes

Remember the EQUANT? Remember PHLOGISTON?

Of course not. Unless you are incredibly nerdy, or perhaps attended one of 4 colleges in the world that still has its students read Ptolemy and Lavoisier, you are completely forgiven for your ignorance. You're not a scientist, right? That kind of specific knowledge is something only they have, right? The reality is, even professional physicists have no recollection of such mistakes of logic commited by their forebears.

Equant and phlogiston theory are two great examples of what happens when theorists are trigger happy to conceive of their theories as fact, or are willing to explain away the empirical data with unreasonable logic in order to avoid recognizing that just perhaps the premises they are working from are false.

There are a myriad of articles just like THIS ONE. Dark Matter is the new Phlogiston. Dark Energy is the new Equant.

The theory of Dark Matter arose out of the necessity to deal with obvious emperical data that depicted a gravitational pull that could not be explained by the existing model of the cosmos. In order to save the model, which worked on many other planes, physicists posited the existence of a material that by all stretches of the imagination cannot exist in the way that we know 'existence'. But they'd rather change metaphysics than physics.

The same goes for Dark Energy. They "are even willing to burn down their old sainted Einstein and revise his theory of gravity, general relativity, to make the cosmic discrepancies go away." This is to say, they are not willing to accept that they are viewing acceleration and motion from a different, relative plane of observation, but are willing to discard the reality of relativity in order to deal with obvious logical fallacies that their further theories present when taken to their natural conclusions. Some of them even recognize that they are betting long odds:

“We are placing a large bet,” Dr. Mountain said, “using our credibility as collateral, that we as a community know what we are doing.”

Even so, I fear the physicists of tomorrow will be standing on the shoulders of midgets. Midgets obsessed with darkness are leading us into another dark age.

Quod Erat Infacetium

We have MATHEMATICAL PROOF that it's not our fault, no matter how hard we tri.


What does it mean?
It means the centroid of attention for any well proportioned figure is located pretty much at its cleavage.
Q.E.I.